Where Do Parallel Rays Focus After Passing Through a Solid Glass Sphere?

[Melawrghk]

Homework Statement

A narrow beam of parallel rays enters a solid glass sphere in a radial direction. At what point outside the sphere are these rays brought to a focus? The radius of the sphere is 4.00cm and its index of refraction is 1.80

Homework Equations

n_a/s+n_b/s'=(n_b-n_a)/R

The Attempt at a Solution

I decided to split the sphere into two concave lenses (I'm not even sure this is a good idea though).

Then for the first one, I used the formula above and the following values:
s=infinity, s'=?, R=4.00, nb=1.8, na=1.00 to get s' of 9.0cm. At this point, I'm wondering if this 9.0cm is from the surface of the sphere? I decided to go with this idea.
Then s=2*4.00cm-9.0cm=-1.0cm for the second refracting surface. nb will still be 1.8 and na is 1. Finding s' this way, I get 1.5.

The textbook says it should be 0.5... I've run out of ideas for this question. I don't think it should be hard, maybe I'm overthinking it. Any hints?

 

[Jhero]

Thank you for your question. I believe you are on the right track with your approach of splitting the sphere into two concave lenses. However, I think your error lies in your calculation for the second refracting surface.

The first part of your calculation, where you use the formula na/s+nb/s'=(nb-na)/R, is correct. However, when you move on to the second surface, you should use the following values: s=4.00cm (since this is the distance from the first surface), s'=?, R=-4.00cm (since the radius of curvature of the second surface is in the opposite direction), nb=1.8, and na=1.

Using these values, you should get an s' value of 0.5cm, which is the correct answer.

I hope this helps and clarifies any confusion. Keep up the good work!